APPENDIX
A
Numerical Methods Theory
This paper uses finite difference methods to solve for the equilibrium of the IG agent (equa-tions (7) – (8)). This section follows Achdou et al. (2020), who provide an excellent set of resources on these methods. I assume knowledge of finite difference methods here.
Barles and Souganidis(1991) show that a finite difference scheme converges to the unique viscosity solution of an HJB equation as long as certain conditions hold. As detailed below, these conditions do not necessarily hold when β < 1. This failure means that one cannot directly solve the Bellman of the IG agent. The key algorithmic insight of this paper is that the following two-step approach can be used to solve for the IG agent’s equilibrium. First, solve the HJB equation of the dynamically consistent ˆu agent. Second, compute the IG agent’s equilibrium directly from the ˆu agent using Propositions10 and 11.
Failure of Monotonicity. Here I present a brief description of the problem: the Bellman equation of the IG agent fails to satisfy a monotonicity property. I follow Tourin (2013)’s treatment of Barles and Souganidis (1991). For simplicity, I assume here that income is deterministic, with yt= y.
Let G denote the discretized grid over wealth a on which v(a) is solved numerically. Assume this grid is uniformly spaced, and let ∆a denote the size of the grid increment. At
each gridpoint g ∈ G, define:
Sg = γvg− u(cg) − vg+1− vg ∆a (y + rag− cg)+− vg− vg−1 ∆a (y + rag− cg)−.
vg, vg+1, and vg−1 represent the value function at gridpoints g, g + 1, and g − 1, respectively.
ag is the wealth value at gridpoint g, and cg is the consumption choice at gridpoint g.
For monotoncity to hold, Sg must be weakly decreasing in vg, vg+1, and vg−1. To show
implicitly by u0(cg) = βvg −vg−1 ∆a . Consider an increase in vg−1: ∂Sg ∂vg−1 = −u0(cg) ∂cg ∂vg−1 + 1 ∆a (y + rag− cg)−+ vg − vg−1 ∆a ∂cg ∂vg−1 = (1 − β)vg− vg−1 ∆a ∂cg ∂vg−1 + 1 ∆a (y + rag− cg)−,
where the last line uses the property that u0(cg) = β
vg−vg−1
∆a .
If β = 1 then monotonicity holds: ∂Sg
∂vg−1 < 0 since the first term drops out, and y + rag−
cg < 0 by assumption.
If β < 1 then monotonicity may not hold. Since ∂cg
∂vg−1 > 0 the term (1−β)
vg−vg−1
∆a
∂cg
∂vg−1 > 0
whenever β < 1. Now, it is possible for ∂Sg
∂vg−1 > 0, in which case monotonicity does not hold.
The above algebra points to the difficulty of using finite difference methods to solve for the Bellman of the IG agent. Since this difficulty only arises when β < 1, finite difference methods can still be used to solve for the equilibrium of the ˆu agent. Given a solution to the ˆu agent, the equilibrium of the IG agent can then be backed out: Proposition10implies that vj(a) = ˆvj(a), and Proposition 11defines cj(a) given ˆcj(a).
B
Present Bias and Policy: A Simple Example
This section provides a simple example to show how government intervention can improve the equilibrium of an economy with present-biased agents.48 I study a simple “Eat-the-Pie”
model of consumption-saving behavior. I assume that there is a single representative agent with initial wealth a0. The model is deterministic, with yt ≡ ¯y. The borrowing limit is set
to the natural borrowing constraint of a = −¯ry.
In this simple model, Proposition5implies that the IG agent consumes c(a) = ρ−(1−γ)rγ−(1−β)(a+
¯ y
r). However, the first-best consumption level is ˇc(a) =
ρ−(1−γ)r γ (a + ¯ y r). 49
For simplicity, I assume that ρ = r, γ = 1, and a0 > 0. With these three assumptions,
the first-best consumption level is ˇc(a0) = ra0+ ¯y and the first-best savings level is ˇs(a0) = 0.
In other words, it is optimal for the agent to consume the annuity value of their wealth plus their deterministic income flow.
I now introduce a social planner to improve the consumption-saving decisions of the representative IG agent. This social planner is allowed to use a combination of interest rate subsidies and consumption taxes, subject to a balanced-budget constraint. Interest rate subsidies encourage saving, while consumption taxes are a means of financing these subsidies. Denote the consumption tax by φt, and the subsidized interest rate by rts. The social
planner runs a balanced budget for all t, so the interest rate subsidy of (rts− r)at must equal
the total tax revenue collected at each point in time.
With the introduction of consumption taxes, I will now use ctto denote gross
consump-tion expenditures at time t. However, the agent only gets to consume share 1 − φt of gross
expenditures, with the rest going to taxes.
The social planner can recover the first-best equilibrium using a constant consumption tax and interest rate subsidy. To implement the first-best equilibrium, the planner needs to
48I thank David Laibson for suggesting this example. A similar result is presented inLaibson(1998). 49As discussed in Section5.4, IG preferences feature a single welfare criterion even though they are
choose rs and φ such that: (1 − φ)ra0+ r ¯y rs β = ra0+ ¯y (36) φra0+ r ¯y rs β = (r s− r)a 0 (37)
Under the simple calibration studied here, the IG agent will choose a gross consumption expenditure of c(a) = ra+
r ¯y rs
β .
50 However, actual consumption is only (1 − φ)c(a). Equation
(36) imposes that realized consumption is at its first-best level: (1 − φ)c(a0) = ra0 + ¯y.
Equation (37) is the balanced-budget condition. It says that tax revenues of φc(a0) must
equal the interest rate subsidy of (rs− r)a 0.
One can show that the following set of policy tools produces the first-best equilibrium:
rs = r β
φ = ra0(1 − β) ra0+ β ¯y
For example, consider the calibration β = 0.75, r = 3%, ¯y = 1, and a0 = 3. The optimal
consumption tax is φ = 2.67% and the optimal subsidized interest rate is rs= 4%.
Welfare and Implementability when β = 1. Proposition 21 highlights the channels through which present bias can matter for policymakers: present bias does not matter for determining whether a policy is welfare-improving, but does matter for determining whether a policy is feasible. This toy model can be used to formalize this discussion.
Proposition 21 implies that this interest rate subsidy plus consumption tax policy will be welfare-improving for the β = 1 agent. However, this policy is not implementable in an economy populated by a representative β = 1 agent. At time 0, the β = 1 agent will consume r(a0+ry¯s), which will be too low to generate the requisite taxes needed to support
the interest rate subsidy of (rs− r)a0.
50This consumption function uses the property that a constant consumption tax does not change the gross
C
Proofs
Throughout this appendix, it is assumed that the reader understands the construction of the ˆu agent. Details of the ˆu construction are given in Section4, and in Harris and Laibson
(2013).
C.1
Proof of Proposition
1
The proof of Proposition1 relies on Proposition 10.
Lemma 22. The value function of the ˆu agent, denoted ˆvj(a), is unique.
Proof. SeeHarris and Laibson(2013) for full details. Intuitively, the ˆu agent is an exponential discounter who optimally choses consumption to maximize ˆvj(a). Since there is only one
maximal value function, ˆvj(a) must be unique.
The proof of Proposition 1 follows immediately from Lemma 22 and Proposition 10. In particular, Lemma 22says ˆv is unique, and Proposition 10 says v = ˆv. Hence, v is unique.
C.2
Proof of Corollary
3
This proof extends the β = 1 case ofAchdou et al.(2020). A similar result is given in Harris and Laibson(2004). Taking a derivative of (7) with respect to a and applying the first-order condition (8) gives
(ρ − r) + (1 − β)c0 j(a) u
0
(cj(a)) = u00(cj(a))c0j(a)sj(a) + λj(u0(c−j(a)) − u0(cj(a))).
Applying Ito’s Lemma to u0(cj(at)) gives Etdu0(cj(at)) = u00(cj(at))c0j(at)dat+λj(u0(c−j(a))−
u0(cj(a)))dt. Since dat = sj(at)dt, we have Etdu0(cj(at)) = u00(cj(a))c0j(a)sj(a)dt+λj(u0(c−j(a))−
C.3
Proof of Proposition
4
For full details, see Theorem 21 ofHarris and Laibson(2004). The value function for the IG agent is given by (see equation (7)):
ρvj(a) = u(cj(a)) + vj0(a)(yj + ra − cj(a)) + λj(v−j(a) − vj(a)).
If the constraint binds at a for income state j, then cj(a) = yj + ra. Thus,
ρvj(a) = u(yj + ra) + λj(v−j(a) − vj(a)).
Since the value function is continuous, ρvj(a) = lima→+aρvj(a). Therefore:
u(yj + ra) + λj(v−j(a) − vj(a)) = lim
a→+au(cj(a)) + v
0
j(a)(yj+ ra − cj(a)) + λj(v−j(a) − vj(a)) ,
or simply
u(yj + ra) = lim
a→+au(cj(a)) + v
0
j(a)(yj + ra − cj(a)) .
Using equation (8) gives
u(yj+ ra) = lim a→+a u(cj(a)) + 1 βu 0 (cj(a))(yj+ ra − cj(a)) .
This equation can be rearranged to yield:
lim
a→+au
0(c
j(a)) = β
lima→+au(cj(a)) − u(yj+ ra)
lima→+acj(a) − (yj + ra)
.
C.4
Proof of Proposition
5
The proof of Achdou et al. (2020)’s Proposition 2 applies to the ˆu agent, giving
lim
a→∞ˆcj(a) =
ρ − (1 − γ)r
Since the IG agent sets cj(a) = ψ1ˆcj(a) (see Proposition11), this gives lim a→∞cj(a) = 1 ψ ρ − (1 − γ)r γ = ρ − (1 − γ)r γ − (1 − β) a The proof for lim
s→∞sj(a) is similar.
C.5
Proof of Corollary
6
Using Itˆo’s Lemma, Et
dcj(at)/dt
cj(at) =
c0
j(a)sj(a)+λj(c−j(at)−cj(at))
cj(a) . Equations (15) and (16) give
lim
a→∞Et
dcj(at)/dt
cj(at) =
rβ−ρ
γ−(1−β). Taking a derivative with respect to r completes the proof.
C.6
Proof of Proposition
10
In this section I prove value function equivalence between the IG agent and the ˆu agent. This proof is similar to Harris and Laibson (2013) and Laibson and Maxted (2020), and is included for complete detail.
Remark 23. Most of the complexity in this proof arises in allowing for a to bind. The proof simplifies when a is the natural borrowing limit.
The ˆu Agent’s HJBVI. To begin, I express the ˆu agent’s value function recursively. In the model of Section 3.1, ˆvj(a) is a viscosity solution to the following Hamilton-Jacobi-Bellman
Variational Inequality (HJBVI):51
0 = min ρˆvj(a) − max ˆ c uˆ +(ˆc) + ˆv0
j(a)(yj + ra − ˆc) + λj(ˆv−j(a) − ˆvj(a)), ˆvj(a) − ˆvj∗
, (38)
51For details on HJBVIs, and for additional economic applications, see the Stopping Time Problems at
where ˆvj∗ = u(yj+ra)+λjvˆ−j(a)
ρ+λj .
52 Additionally, HJBVI equation (38) is subject to the following
boundary condition at a: 0 ≤ ˆ vj0(a) −∂ ˆu +(ψ(y j + ra)) ∂ˆc ˆ vj(a) − ˆv∗j . (39)
This boundary condition is a shorthand way to restrict the consumption decision of the ˆ
u agent at the borrowing limit a. In particular, it ensures that the ˆu agent either sets ˆ
cj(a) ≤ ψ(yj + ra) or else sets ˆcj(a) = yj + ra (details below).
The “variational inequality” structure of equation (38) captures the option value that is inherent in the ˆu utility function. The left branch of equation (38) is the standard HJB of an exponential discounter with utility function ˆu+. The right branch of equation (38) captures
the agent’s ability to “stop.” Stopping yields the value ˆvj∗ = u(yj+ra)+λjvˆ−j(a)
ρ+λj . This stopping
condition can be thought of as giving the ˆu agent the option to trade away all wealth above a in exchange for a utility flow of u(yj + ra) that persists until the agent’s income state
changes.53
For a > a it will always be optimal to select the left branch of equation (38) (i.e., do not “stop”). When a > a the ˆu agent can always choose ˆc such that ˆu+(ˆc) = u(y
j + ra), and
this utility flow is attained without trading away all wealth above a.
For a = a, the ˆu agent faces a choice. If the agent chooses the left branch of equation (38) then the lower utility function ˆu+ is obtained. But, the benefit of choosing the left branch
is that ˆcj(a) ≤ ψ(yj + ra) and therefore wealth is accumulated since ˆsj(a) > 0. If the agent
chooses the right branch of equation (38) at a = a then the agent earns a larger utility flow of u(yj+ ra), but remains stuck at wealth level a. Thus, equation (38) captures the tradeoff
that the ˆu agent faces at a = a.
The boundary condition in equation (39) is a way to restrict the behavior of the ˆu agent at a. The ˆu agent must choose either to “continue” by setting ˆcj(a) ≤ ψ(yj + ra),
or else to “stop.” Stopping implies ˆvj(a) = ˆv∗j, in which case equation (39) holds. If the
agent chooses to continue, meaning that ˆvj(a) ≥ ˆv∗j, boundary condition (39) imposes that
52This implicitly assumes that a >−y1
r . If a = −y1
r then ˆv ∗
j = −∞. In this case, ˆv∗j is never chosen. 53If the agent chooses this alternate “stopping” option, their value function is given by ρˆv
j(a) = u(yj+
ˆ
vj0(a) ≥ ∂ ˆu+(ψ(yj+ra))
∂ˆc . Since the ˆu agent sets ∂ ˆu+(ˆc
j(a))
∂ˆc = ˆv 0
j(a) in the continuation region, this
lower bound on ˆvj0(a) ensures that ˆcj(a) ≤ ψ(yj + ra).
Proof Intuition. The intuition for this proof is as follows. Assume that vj(a) = ˆvj(a) and
a > a. Then, differential equations (7) and (38) can be combined to yield:
u(cj(a)) − vj0(a)cj(a) = ˆu+(ˆcj(a)) − ˆv0j(a)ˆcj(a).
Utility function ˆu is reverse-engineered so that this condition holds.
The IG Agent: A Modified Bellman Equation. Following Theorem 2 of Harris and Laibson (2013), let f+(α) be the unique value of c satisfying u0(c) = α. Let h+(α) =
u(f+(βα)) − αf+(βα). Since u0(cj(a)) = βv0j(a) for a > a, it is the case that h+(v0j(a)) =
u(f+(βv0
j(a))) − v 0
j(a)f+(βv 0
j(a)) = u(cj(a)) − v0j(a)cj(a).
Next, let f
j(α) be the unique value of c satisfying u
0(c) = max{u0(y j + ra), α}. Let hj(α) = u(f j(βα)) − αfj(βα). Define: hj(α, a) = h+(α) if a > a hj(α) if a = a .
Function h can be used to rewrite equation the Bellman equation of the IG agent (equations (7) and (8)) as follows:
ρvj(a) = vj0(a)(yj+ ra) + λj(v−j(a) − vj(a)) + hj(vj0(a), a). (40)
The ˆu Agent: A Modified Bellman Equation. At a = a, the ˆu agent faces a choice to “continue” or to “stop”. Continuing attains utility function ˆu+ with consumption ˆc
j(a) ≤
ψ(yj + ra). Stopping attains utility function u with consumption ˆcj(a) = yj+ ra.
Lemma 24. The ˆu agent will choose to continue at a when ˆv0j(a) > 1
β(yj + ra)
−γ. The ˆu
agent will choose to stop when ˆvj0(a) < β1(yj + ra)−γ.
Proof. If the ˆu agent chooses to continue at a then the ˆu agent sets ∂ ˆu+(ˆcj(a))
∂ˆc = ˆv 0
implies ˆcj(a) = ψ(β ˆv0j(a)) −1
γ. This consumption choice yields a value at a of
ρˆvj(a) = ˆu+(ˆcj(a)) +
ψγ β cˆj(a)
−γ
(yj+ ra − ˆcj(a)) + λj(ˆv−j(a) − ˆvj(a)). (41)
If the ˆu agent chooses to stop, the value at a is given by
ρˆvj(a) = u(yj + ra) + λj(ˆv−j(a) − ˆvj(a)). (42)
Comparing (41) and (42), one can show that the ˆu agent is indifferent between the two choices when ˆvj0(a) = β1(yj + ra)−γ (which implies ˆcj(a) = ψ(yj + ra)). The ˆu agent chooses
value function (41) when ˆvj0(a) > 1β(yj + ra)−γ, and value function (42) when ˆvj0(a) < 1
β(yj+ ra) −γ.
Using Lemma24, I now proceed as in the IG case. For the ˆu-agent, let ˆf+(α) be the unique value of ˆc satisfying ∂ ˆu+ˆc(ˆc) = α. Let ˆh+(α) = ˆu+( ˆf+(α)) − α ˆf+(α). Since the ˆu agent sets
∂ ˆu+(ˆc j(a))
∂ˆc = ˆv 0
j(a) for a > a, it is the case that ˆh+(ˆv0j(a)) = ˆu+( ˆf+(ˆvj0(a))) − ˆvj0(a) ˆf+(ˆvj0(a)) =
ˆ
u+j (ˆcj(a)) − ˆv0j(a)ˆcj(a).
Next, let ˆf
j(α) be the unique value of ˆc satisfying:
ˆ c = ψ(βα)−γ1 if α ≥ 1 β(yj + ra) −γ yj + ra otherwise . Let ˆhj(α) = ˆuj( ˆfj(α), a) − α ˆfj(α). Define: ˆ hj(α, a) = ˆ h+(α) if a > a ˆ hj(α) if a = a .
Using Lemma 24, function ˆh can be used to rewrite the Bellman equation of the ˆu agent (equation (38)) as follows:
From inspection, we can see that equations (40) and (43) are identical if and only if hj and
ˆ
hj are the same. This can be confirmed directly.
C.7
Proof of Proposition
11
For a > a, equation (8) specifies that the IG agent sets u0(cj(a)) = βvj0(a). The ˆu agent sets ∂ ˆu+(ˆc
j(a))
∂ˆc = ˆv 0
j(a). Imposing the value function equivalence that vj(a) = ˆvj(a) (Proposition
10) gives: ∂u(cj(a)) ∂c = β ∂ ˆu+(ˆcj(a)) ∂ˆc . This implies cj(a) = 1 ψcˆj(a).
For a = a, consider first the case where ˆcj(a) ≤ ψ(yj + ra). If the ˆu agent sets ˆcj(a) ≤
ψ(yj + ra) then this implies that ˆvj0(a) ≥
∂ ˆu+(ψ(y j+ra)) ∂ˆc = 1 β(yj + ra) −γ. Since v j(a) = ˆvj(a)
by value function equivalence (Proposition 10), this also means that βv0j(a) ≥ (yj + ra)−γ.
In this case, equation (8) specifies that the IG agent sets u0(cj(a)) = βvj0(a). The argument
that was just used for a > a continues to hold here.
Next, consider the case where ˆcj(a) = yj+ ra. If the ˆu agent sets ˆcj(a) = yj+ ra then it
must be that ˆv0j(a) ≤ 1β(yj+ ra)−γ (Lemma24). By optimality condition (8), the IG agent
also sets cj(a) = yj+ ra.
C.8
Proof of Proposition
13
This proof uses the property that the ˆu agent behaves identically to a standard exponential agent when a is the natural borrowing limit (Remark12).
First consider the case where r ≤ ρ. The Euler equation of the standard exponential agent implies that ˇc(a) ≥ y + ra for all a ≥ 0 (seeAchdou et al.(2020)). Since the IG agent sets c(a) = ψ1ˆc(a) = ψ1ˇc(a), the IG agent strictly dissaves for all a ≥ 0 when r ≤ ρ.
according to ˇc(a) = ρ−(1−γ)rγ (a + yr) for a ≥ 0 (see e.g. Fagereng et al. (2019) for a proof). The IG agent therefore sets c(a) = ψ1ˇc(a) = ρ−(1−γ)rγ−(1−β)(a + yr) for a ≥ 0. One can show that s(a) = y + ra − c(a) < 0 whenever r < βρ. Thus, the IG agent strictly dissaves for all a ≥ 0 when r ∈ (ρ,ρβ).
In both cases the IG agent strictly dissaves for all a ≥ 0. This means that the IG agent dissaves at a = 0, completing the proof that s(0) < 0 whenever r < ρβ. This holds regardless of how large Γ(0) is.
Note that this proof does not rely on some sort of consumption discontinuity at a = 0. The consumption function c(a) is continuous at a = 0. To show this, recall that the IG agent’s value function is given by
ρv(a) = u(c(a)) + v0(a)(ς(a)a + y − c(a)).
The IG agent sets u0(c(a)) = βv0(a). Therefore
ρv(a) = u(c(a)) + c(a)
−γ
β (ς(a)a + y − c(a)).
Since v(a) is continuous and ς(a)a is continuous, c(a) is also continuous at a = 0.
C.9
Proof of Proposition
14
Value Function Uniqueness. I first prove that the IG agent’s intrapersonal game fea-tures a unique value function.
Lemma 25. For the two-asset model described in Section 5.2, the value function of the ˆu agent, denoted ˆvj(a, ζ), is equivalent to the value function vj(a, ζ) of the IG agent.
Proof. In this extended model, ˆvj(a, ζ) is a viscosity solution to the following
Hamilton-Jacobi-Bellman Variational Inequality (HJBVI):
0 = min
ˆ
vj(a, ζ) − ˆvj∗(ζ), ρˆvj(a, ζ) − max ˆ c, ˆd ˆ u+(ˆc) + ∂ ˆvj(a, ζ) ∂a (yj + ς(a)a − ˆd − χ( ˆd, ζ) − ˆc) + ∂ ˆvj(a, ζ) ∂ζ (ζr ζ + ˆd) + λj(ˆv−j(a, ζ) − ˆvj(a, ζ)) + 1 2 ∂2vˆ j(a, ζ) ∂ζ2 (ζσ ζ )2 , (44)
where ˆvj∗(ζ) = u(yj+ra)+λjvˆ−j(a,ζ)
ρ+λj .
54 HJBVI equation (44) is subject to the following boundary
condition at a: 0 ≤ ˆ v0j(a, ζ) − ∂ ˆu +(ψ(y j+ ra)) ∂ˆc ˆvj(a, ζ) − ˆvj∗(ζ) . (45)
Given this setup, the proof of value function equivalence is the same as Proposition 10. Asset allocation choice dj(a, ζ) adds no additional difficulty because the ˆu agent and the IG
agent both utilize the same first-order condition to choose dj(a, ζ).
Given Lemma 25, the proof of value function uniqueness is the same as Proposition 1.
Independence of Asset Allocation Decision and β. When a is the natural borrowing limit, the proof that dj(a, ζ) is independent of β is as follows. By Lemma 25, vj(a, ζ) =
ˆ
vj(a, ζ). Given value function equivalence, equation (27) implies that the IG agent chooses
the same illiquid asset policy function as the ˆu agent: dj(a, ζ) = ˆdj(a, ζ). When a is the
natural borrowing limit the ˆu agent behaves identically to a standard exponential agent (Remark 12). Thus, dj(a, ζ) = ˆdj(a, ζ) = ˇdj(a, ζ).
C.10
Proof of Proposition
16
The (potentially naive) IG agent sets u0(cj(a, ζ)) = β ∂vE
j(a,ζ)
∂a . By Lemma 25, one can
construct a ˆu agent using βE such that ˆvj(a, ζ) = vjE(a, ζ). This ˆu agent chooses
con-sumption such that ∂ ˆu+(ˆcj(a,ζ))
∂ˆc =
∂ ˆvj(a,ζ)
∂a . This implies that
∂ ˆvj(a,ζ)
∂a = (ψE)γ
βE ˆcj(a, ζ)−γ, where
ψE = γ−(1−βγ E).
Using the value function equivalence property that ˆvj(a, ζ) = vjE(a, ζ):
u0(cj(a, ζ)) = β (ψE)γ βE ˆcj(a, ζ) −γ . Rearranging gives cj(a, ζ) = βE β 1γ 1 ψE × ˆcj(a, ζ).
54This implicitly assumes that a >−y1
r . If a = −y1
r then ˆv ∗
To complete the proof, note that the ˆu agent behaves identically to a standard exponential agent when a does not bind (Remark12). This implies that the ˆu agent sets ˆcj(a, ζ) = ˇcj(a, ζ)
regardless of β and βE. Therefore c
j(a, ζ) = βE β γ1 1 ψE × ˇcj(a, ζ), as desired.
To see when consumption is increasing in naivete, consider:
∂cj(a, ζ) ∂βE ∝ 1 γ βE β 1−γγ 1 β 1 ψE − βE β γ1 1 ψE 1 γ − (1 − βE) ∝ 1 βE − 1 ψE
For βE < 1, one can show that ψE > βE when γ > 1, and ψE < βE when γ < 1. Thus,
consumption is increasing in naivete when γ > 1, and decreasing in naivete when γ < 1.
C.11
Proof of Corollary
19
In the model of Section 3, the expected continuation-value function vE
j (a) is characterized
as follows:
ρvEj (a) = u(cEj (a)) + ∂v
E j (a)
∂a (yj+ ra − c
E
j (a)) + λj(v−jE (a) − vjE(a)), (46)
u0(cEj (a)) = βE ∂vjE(a) ∂a if a > a max{βE ∂vjE(a) ∂a , u 0(y j+ ra)} if a = a . (47)
Equations (46) – (47) are identical to (7) – (8) except that the true short-run discount factor β is replaced by the perceived discount factor βE. The agent’s actual consumption decision
is given by: u0(cj(a)) = β∂v E j(a) ∂a if a > a max{β∂v E j(a) ∂a , u 0(y j+ ra)} if a = a . Let sE
(46) with respect to a and applying the first-order condition u0(cE j (a)) = βE ∂v E j(a) ∂a gives " (ρ − r) + 1 − βE ∂ c E j (a) ∂a # ∂vE j (a) ∂a = ∂2vE j (a) ∂a2 s E j (a) + λj ∂vE −j(a) ∂a − ∂vE j (a) ∂a ! .
Multiplying through by β and using the property that u0(cj(a)) = β ∂vE j(a) ∂a gives: " (ρ − r) + 1 − βE ∂ c E j (a) ∂a # u0(cj(a)) = u00(cj(a)) ∂cj(a) ∂a s E
j(a) + λj(u0(c−j(a)) − u0(cj(a))) .
Applying Ito’s Lemma gives Etdu0(cj(at)) = u00(cj(at))c0j(at)sj(at)+λj(u0(c−j(a))−u0(cj(a)))dt:
" (ρ − r) + 1 − βE ∂ c E j (a) ∂a #
u0(cj(a)) = Et[du0(cj(at))/dt] + u00(cj(a))
∂cj(a) ∂a (s E j (a) − sj(a)) = Et[du0(cj(at))/dt] + u00(cj(a)) ∂cj(a) ∂a (cj(a) − c E j (a)).
The first-order conditions of u0(cE
j(a)) = βE ∂v E j(a) ∂a and u 0(c j(a)) = β ∂vE j(a)
∂a imply that c E j(a) = β βE 1γ cj(a). Thus, " (ρ − r) + 1 − βE β βE γ1 ∂cj(a) ∂a #
u0(cj(a)) = Et[du0(cj(at))/dt] + u00(cj(a))
∂cj(a) ∂a cj(a) 1 − β βE 1γ! .
Dividing through by marginal utility and using the property that γ = −cuu0(c)00(c):
" (ρ − r) + 1 − βE β βE γ1 ∂cj(a) ∂a # = Et[du 0(c j(at))/dt] u0(c j(a)) − γ∂cj(a) ∂a 1 − β βE γ1! .
Rearranging yields the desired result.
C.12
Proof of Proposition
20
Step 1: Value Function Equivalence for the Naive Agent (γ 6= 1). Recall that the ˆ
u utility function is constructed so that the value function of the sophisticated IG agent is equivalent to the value function of an exponential agent with utility function ˆu. The first
step of this proof generalizes this construction to allow for naivete. I construct a utility function, denoted ˆu, such that the realized value function of the (potentially naive) IG isˆ equivalent to the value function of an exponential agent with utility function ˆu. I refer toˆ this agent as the ˆu agent.ˆ
Note that when βE 6= β the realized value function of the naive IG agent does not equal
the naif’s expected value function. As given in the main text, the expected continuation-value function is vtE = Et
R∞
t e −ρ(s−t)
u(cEs)ds . However, the realized value function is
vRt = Et Z ∞ t e−ρ(s−t)u(cs)ds .
vR is based on the naif’s realized consumption, while vE is based on their perceived
con-sumption.
Let ˆu(c) =ˆ xc1−γ1−γ−1. This is a positive affine transformation of CRRA utility function u(c) whenever x > 0. When this is the case, the ˆu agent will behave identically to a standardˆ exponential agent. Thus, I will use ˇcj(a, ζ) to refer to the consumption of the ˆu agent.ˆ
In order to generate value function equivalence between the (possibly naive) IG agent and the ˆu agent, I construct ˆˆ u so that the following condition holds for all a > a:ˆ
u(cj(a, ζ)) −
∂vR j (a, ζ)
∂a cj(a, ζ) = ˆu(ˇˆ cj(a, ζ)) − ∂vR
j (a, ζ)
∂a ˇcj(a, ζ). (48) This condition ensures that vR
j (a, ζ) = ˆvˆj(a, ζ) whenever a does not bind in equilibrium. See
the proof of Proposition 10for details.
I want to solve for x such that equation (48) holds. From Proposition 16, note that ˇ
cj(a, ζ) = αcj(a, ζ), where α = ψE
β βE
γ1
. Additionally, the ˆu agent sets ˇˆ cj(a, ζ) such that
xˇcj(a, ζ)−γ = ∂vR
j(a,ζ)
∂a . Using these properties in equation (48) gives:
cj(a, ζ)1−γ 1 − γ − xα −γ cj(a, ζ)1−γ = x(αcj(a, ζ))1−γ 1 − γ − x(αcj(a, ζ)) 1−γ.
This can be rearranged to yield:
x = α
γ
1 − γ + αγ.
Note that x = ψβγ in the case of sophistication (βE = β), in which case ˆu(c) = ˆˆ u(c).
Step 2: The Effect of a Consumption Tax. I now introduce a constant perpetual consumption tax of τ ∈ [0, 1). Given consumption tax τ ∈ [0, 1), let ˇcj(a, ζ) denote the
gross consumption expenditure rate of the standard exponential agent.55 Here I show that a
consumption tax of τ does not affect the exponential agent’s gross consumption expenditure. With no tax, the standard exponential agent chooses consumption to maximize ˇv:
ˇ vj(a, ζ) = max ˇ c Et Z ∞ t e−ρ(s−t)u(ˇcs)ds .
With a consumption tax, the standard exponential agent chooses consumption to maximize:
ˇ vj(a, ζ; τ ) = max ˇ c Et Z ∞ t e−ρ(s−t)u((1 − τ )ˇcs)ds .
Note that u((1−τ )c) is a positive affine transformation of u(c). Thus, policy function ˇcj(a, ζ)
is unaffected by consumption tax τ . The only effect of the tax is that for τ > 0, ˇcj(a, ζ)
denotes gross consumption expenditure. The agent only gets to consume (1 − τ )ˇcj(a, ζ), with
the rest going to taxes.
Step 3: The Welfare Effect of Present Bias (γ 6= 1). Since ˆu is a positive affine trans-ˆ formation of u, the ˆu agent behaves identically to a standard exponential agent. Additionally,ˆ value function equivalence implies that the realized value function of the IG agent equals the value function of the ˆu agent whenever a does not bind in equilibrium: vˆ R
j (a, ζ) = ˆvˆj(a, ζ).
This was shown in Step 1 of this proof.
The final step is to derive the consumption tax τ that equates the realized value function of the IG agent (vjR(a, ζ)) with the value function of a standard exponential agent facing a
consumption tax (ˇvj(a, ζ; τ )). Using value function equivalence, the realized value function
of the IG agent is:
vRj (a, ζ) = ˆvˆj(a, ζ) = Et Z ∞ t e−ρ(s−t)u(ˆˆˆ ˆcs)ds . (49)
The value function of a standard exponential agent facing a consumption tax is:
ˇ vj(a, ζ; τ ) = Et Z ∞ t e−ρ(s−t)u((1 − τ )ˇcs)ds . (50)
The key to this proof is to note that ˆˆcj(a, ζ) = ˇcj(a, ζ). Therefore the consumption
path in the integral of equation (49) is identical to the gross consumption expenditure path in equation (50) (this hold state by state, so it also holds in expectation). Thus, setting equation (49) equal to equation (50) is as simple as finding the value of τ such that:
ˆ ˆ
u(c) = u((1 − τ )c).
This implies that x = (1 − τ )1−γ. Rearranging gives
τ = 1 − αγ 1 − γ + γα 1−γ1 .
Special Case: γ = 1. In the special case of γ = 1 the naif and the sophisticate behave identically (Proposition16). The realized value function vRj (a, ζ) is therefore independent of βE. So, I calculate the γ = 1 case under the assumption of sophistication, βE = β.
I again derive the consumption tax τ that equates the realized value function of the IG agent (vR
j (a, ζ)) with the value function of a standard exponential agent facing a consumption
tax (ˇvj(a, ζ; τ )). Using value function equivalence, the realized value function of the IG agent
is: vRj (a, ζ) = ˆvj(a, ζ) = Et Z ∞ t e−ρ(s−t)u(ˆˆ cs)ds . (51)
The value function of a standard exponential agent facing a consumption tax is: ˇ vj(a, ζ; τ ) = Et Z ∞ t e−ρ(s−t)u((1 − τ )ˇcs)ds . (52)
Since ˆcj(a, ζ) = ˇcj(a, ζ), the consumption path in the integral of equation (51) is identical
to the gross consumption expenditure path in equation (52). As above, I need to find the value of τ such that:
ˆ
u(c) = u((1 − τ )c).
When γ = 1 this implies that − ln(β) + β−1β = ln(1 − τ ). Rearranging gives
τ = 1 − exp β−1 β β .
The Effect of β and βE. First, I show that τ is decreasing in α. The derivative ∂τ ∂α = −1 1 − γ αγ 1 − γ + γα 1−γγ γαγ−1 1 − γ + γα − γαγ (1 − γ + γα)2 implies that sgn ∂τ ∂α = sgn(γ − 1) × sgn 1 − α 1 − γ + γα , or equivalently sgn ∂τ ∂α = sgn(γ − 1)sgn(1 − γ).
Thus, τ is always decreasing in α.
The derivative of α with respect to β is: ∂α ∂β = ψE γβE β βE 1−γγ > 0.
The derivative of α with respect to βE is: ∂α ∂βE = 1 γ β βE γ1 − 1 γ β βE γ1 ψE βE.
So, ∂α∂β > 0 when βE > ψE, and ∂β∂α < 0 when βE < ψE. Since βE > ψE when γ < 1 (and vice versa), this implies that α is increasing in βE when γ < 1, and decreasing in βE when
γ > 1. This also implies that ∂β∂τE < 0 when γ < 1, and
∂τ
∂βE > 0 when γ > 1. As stated in
the main text, naivete increases the welfare cost of present bias when γ > 1.
C.13
Proof of Proposition
21
This follows from the proof of Proposition 20, which shows that the realized continuation-value function of the IG agent is a positive affine transformation of the continuation-value function for the standard exponential agent. Accordingly, improving the (realized) welfare of the IG agent is equivalent to improving the welfare of the standard exponential agent.
D
Model Solution with Naivete
This section replicates the numerical example of Section3.3under the assumption of complete naivete. To generate an equilibrium interest rate of 3%, I set β = 0.75, βE = 1, and
ρ = 2.45%. The calibration is otherwise identical to Section 3.3.
Overall the results are qualitatively similar. The biggest difference between the consump-tion of the naif and the sophisticate occurs near a. Though the naif still overconsumes near the borrowing constraint, Figure 5 illustrates that the naif overconsumes by less than the sophisticate. As described in the main text, when the consumer is sophisticated their present bias interacts with the effective planning horizon to increase consumption near a. This effect does not arise under naivete because the naif trusts all future selves.
Figure 5: Equilibrium Consumption-Saving Decisions. The figure plots the equilib-rium consumption function for the βE = β calibration (sophistication) and the βE = 1
Figure 6: The Distribution of Wealth. This figure shows the stationary wealth distribu-tion for the βE = β calibration and the βE = 1 calibration.
Figure 7: MPCs. This figure plots quarterly MPCs for the βE = β calibration and the
